MinT - Best Known (86−56, 86, s)-Nets in Base 3

Best Known (86−56, 86, s)-Nets in Base 3

(86−56, 86, 37)-Net over F₃ — Constructive and digital

Digital (30, 86, 37)-net over F₃, using

t-expansion [i] based on digital (27, 86, 37)-net over F₃, using
- net from sequence [i] based on digital (27, 36)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(86−56, 86, 42)-Net over F₃ — Digital

Digital (30, 86, 42)-net over F₃, using

t-expansion [i] based on digital (29, 86, 42)-net over F₃, using
- net from sequence [i] based on digital (29, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 29 and N(F) ≥ 42, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(86−56, 86, 101)-Net in Base 3 — Upper bound on s

There is no (30, 86, 102)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁸⁶, 102, S₃, 56), but
- the linear programming bound shows that MÂ â‰¥ 229 126665 420746 420271 037058 567775 156188 587579 935531 / 1773 341440 > 3⁸⁶ [i]